Methods for Multi-Loop Identification of
Visual and Neuromuscular Pilot Responses
Mario Olivari, Student Member, IEEE, Frank M. Nieuwenhuizen, Joost Venrooij, Student
Member, IEEE,
Heinrich H. B¨ulthoff, Member, IEEE, and Lorenzo Pollini, Member, IEEE
Abstract
In this paper, identification methods are proposed to estimate the neuromuscular and visual responses of a multi-loop pilot model. A conventional and widely used technique for simultaneous identification of the neuromuscular and visual systems makes use of cross-spectral density estimates. This paper shows that this technique requires a specific non-interference hypothesis, often implicitly assumed, that may be difficult to meet during actual experimental designs. A mathematical justification of the necessity of the non-interference hypothesis is given. Furthermore, two methods are proposed that do not have the same limitations. The first method is based on ARX models, whereas the second one combines cross-spectral estimators with interpolation in the frequency domain. The two identification methods are validated by offline simulations and contrasted to the classic method. The results reveal that the classic method fails when the non-interference hypothesis is not fulfilled; on the contrary, the two proposed techniques give reliable estimates. Finally, the three identification methods are applied to experimental data from a closed-loop control task with pilots. The two proposed techniques give comparable estimates, different from those obtained by the classic method. The differences match those found with the simulations. Thus, the two identification methods provide a good alternative to the classic method and make it possible to simultaneously estimate a human’s neuromuscular and visual response in cases where the classic method fails.
Index Terms
M. Olivari is with the Department of Human Perception, Cognition and Action, Max Planck Institute for Biological Cybernetics, 72076 T¨ubingen, Germany, and also with Dipartimento di Ingegneria dell’Informazione, University of Pisa, 56122 Pisa, Italy (e-mail: mario.olivari@tuebingen.mpg.de)
F. M. Nieuwenhuizen is with the Department of Human Perception, Cognition and Action, Max Planck Institute for Biological Cybernetics, 72076 T¨ubingen, Germany (e-mail: frank.nieuwenhuizen@tuebingen.mpg.de).
J. Venrooij is with the Department of Human Perception, Cognition and Action, Max Planck Institute for Biological Cybernetics, 72076 T¨ubingen, Germany, and also with the Faculty of Mechanical, Maritime and Materials Engineering, Delft University of Technology (TU Delft), 2628CD Delft, The Netherlands (e-mail: joost.venrooij@tuebingen.mpg.de).
L. Pollini is with Dipartimento di Ingegneria dell’Informazione, University of Pisa, 56122 Pisa, Italy (e-mail: lorenzo.pollini@unipi.it). H. H. B¨ulthoff is with the Department of Human Perception, Cognition and Action, Max Planck Institute for Biological Cybernetics, 72076 T¨ubingen, Germany, and also with the Department of Brain and Cognitive Engineering, Korea University, Seoul 136-7601, Korea (e-mail: heinrich.buelthoff@tuebingen.mpg.de).
Pilot identification, neuromuscular system, haptic feedback, tracking task.
I. INTRODUCTION
Novel pilot support systems include haptic aids that help pilots perform a control task by providing additional tactile information on the control device [1]–[3]. Haptic aids have been widely employed in several contexts. Examples are remote control with the master-slave paradigm [1], [4], force feedback in servo-actuated systems [5], and robotic assisted surgery [6]. In these contexts the haptic channel is used to let the operator feel variations and/or discontinuities in the environment, like hitting or approaching an obstacle, encountering unexpected loads, or interacting with surfaces (i.e. tissues) with different softness. This knowledge eventually increases performance and situational awareness. Other studies investigated the use of haptic feedback for generating synthetic warnings. So-called stick shakers are currently used in many modern aircraft for alerting pilots to a potential stall [7], [8].
In recent years, there has been considerable interest in designing haptic aids as guidance systems. Such aids
continuously provide a haptic feedback on the control device that suggests a possible right control action to the
pilot. Examples are a haptic gas pedal that assists the driver in car following [9], a haptic steering wheel for lane-keeping [3], [10], and a haptic flight director that assists pilots in following a glide slope [11], [12]. The pilot can either follow the continuous haptic feedback, or override it in case of discordance. It has been shown that pilots quickly get acquainted with this type of haptic feedback, learn how to exploit it to increase their performance, and adapt their responses to the specific type of haptic aid [10], [13].
Due to the continuous nature of the tactile feedback, one can assume that pilots adopt a time-invariant behavior. Since humans can adapt their responses over a large range [13], [24], an effective design of haptic aid requires accurate knowledge of pilot adaptation to the provided continuous haptic feedback. This papers seeks to address how to model and estimate the pilot responses in control tasks with continuous haptic feedback.
Quasi-linear models are commonly used to describe pilot behavior in various control tasks [14]–[16]. These models represent pilot behavior with linear describing functions combined with a remnant signal to account for non-linearities. When haptic aids are used and tactile information is provided in addition to presenting visual information, the tactile perception channel needs to be modeled as an additional input to the pilot. A multi-loop quasi-linear model that meets this requirement was developed by van Paassen et al. [19], [20]. This model is characterized by two control loops, one describing the visual perception channel and one describing the tactile information processed by the neuromuscular system. Identifying the neuromuscular and visual responses would provide quantitative insights into the pilot adaptation to the haptic feedback.
To identify pilot visual and neuromuscular responses during a closed-loop control task, identification techniques are needed that account for the multi-loop nature of the task. A commonly used technique is based on the Cross Spectral Density (CSD) analysis [21]. This technique has been initially applied to neuromuscular system estimation for tasks in which pilot visual feedback is not involved and the pilot is essentially a single-input single-output system [22]. In [13], [23] the technique was extended to estimate neuromuscular system in control tasks with visual feedback. Two mutually uncorrelated forcing functions, a visual reference trajectory and a force
V IS CD ADM−1 CE +− xtar e xCE n +Fhuman xCD Fvis Fadm Fdist P ilot Fhapt +− +++
Fig. 1. Compensatory tracking task, in which a pilot controls the Controlled Element (CE) through a Control Device (CD). The pilot is described with a visual (VIS) and a neuromuscular response (the inverse of the admittance ADM), combined with the remnant signal n. The dashed arrows indicate signals that can not be measured, while the large shaded arrows represent the known forcing functions needed for identification. All other signals can be measured.
disturbance, were inserted in the feedback loop to estimate the neuromuscular and visual dynamics. The contribution of force disturbance in the visual error was implicitly assumed to be negligible, i.e., the force disturbance used for identification was assumed not to interfere with the visual feedback. Unfortunately, this non-interference assumption is not always verified [24]. As a sufficiently high Signal-to-Noise Ratio (SNR) is required for the identification procedure, the power of the force disturbance cannot be decreased such that its effect on the visual feedback is not actually perceived by the pilot.
This paper shows that the commonly used CSD method may provide biased estimates when the non-interference assumption is not fulfilled. Two novel identification methods are presented, which also produce reliable estimates when the force disturbance influences the visual error. In the first method (ARX-method), a multi-loop Auto-Regressive model with eXogenous inputs (ARX) is used to fit the measured data in the time domain. A similar identification procedure was successfully applied for a different control task [25]. The second method (CSD-ML method) makes use of the Cross Spectral Density analysis like the conventional method, but does not require the non-interference assumption. Numerical simulations and human in-the-loop experiment are performed to provide supportive evidence of the reliability of the two novel methods.
The paper is structured as follows: Section II presents a model that describes pilots in a tracking task with continuous haptic aids. Sections III and IV illustrate the theoretical basis of the three adopted identification methods, followed by a set of off-line Monte Carlo simulations in section V. Subsequently, section VI shows an experimental validation of the proposed identification methods. Finally, conclusions are drawn.
II. PILOTMODEL ANDCONTROLTASKDESCRIPTION
This paper focuses on human control behavior in a compensatory tracking task. In this task, pilots are asked
to follow a reference signal by compensating for a tracking error e. When haptic aids are used, additional tactile
information is provided to the pilot for achieving the control task. To assess the influence of this second input on the pilot, the visual and tactile perception channels need to be modeled separately.
A multi-loop model that meets these requirements was developed by van Paassen et al. [13], [19], see Fig. 1. The human operator is described by two control loops, the outer describing the visual response, whereas the inner loop describes the neuromuscular response. The neuromuscular response is represented by the inverse of the arm admittance ADM, defined as the dynamic relationship between the force acting on the arm and the position of the arm [13]. The visual system VIS and the neuromuscular response ADM both contribute to the control force Fhuman. As common in the context of pilot dynamics identification with continuous haptic aids [9], [13], the visual
and the neuromuscular responses are assumed to be produced by linear and time-invariant systems. Nonlinearities
in the pilot response are accounted for by the remnant signaln [15].
The haptic forceFhaptrepresents any additional force feedback provided by the haptic aid. In the field of guidance
haptic aids, the haptic system is generally designed as a standard compensator, which continuously provides forces that aim at achieving a certain control task [24], [27]. Referring to the compensatory tracking task shown in Fig. 1,
the haptic compensator is commonly designed to regulate the tracking errore to zero. Throughout this paper, we will
consider the dynamics of the haptic compensator equal to zero for the purposes of developing the pilot identification techniques. This simplification does not affect the generality of the developed identification methods, since they are easily extensible to the case of non-zero haptic aid dynamics.
III. CONVENTIONALIDENTIFICATIONTECHNIQUE
This section describes the identification technique, based on cross-spectral analysis, that is commonly used to estimate the pilot responses of the model shown in Fig. 1. To simultaneously estimate the admittance and the
visual response, two deterministic forcing functions need to be inserted into the loop: the target position xtar,
which represents the reference trajectory that pilot has to track, and the force disturbanceFdist, which represents
a continuous force disturbance that the pilot can feel on the control device.
The forcing functions have to be tuned with care, as the presence of human in the control loop poses constraints on their power content. The amount of power and the bandwidth should not be too high, to prevent changes in the human control strategy [21], [28], [29]. At the same time, they should not be too low in order to ensure an accurate estimate. In order to satisfy both conflicting constraints simultaneously, a trade-off has to be found.
A. Forcing Functions Design
The two forcing functionsxtar andFdist were designed as multisine signals [30]:
xtar(t) = Nt X j=1 Tjsin(2πfTjt + ψTj) (1) Fdist(t) = Nd X j=1 Djsin(2πfDjt + ψDj) . (2)
Each frequencyfDj andfTj was chosen as an integer multiple of the base frequency, which is defined as the inverse
of the measurement timeT of the forcing function. In this case, all measurements lasted 81.92 s, corresponding to
10−1 100 101 10−4 10−3 10−2 10−1 100
Power Spectral Density
Frequency [Hz] F dist [N 2/Hz] x tar [deg 2/Hz]
(a) Power Spectral Densities.
0 20 40 60 80 −5 0 5 Fdist [N] 0 20 40 60 80 −5 0 5 Time [s] xtar [deg] (b) Sample realizations. Fig. 2. Forcing signals xtarand Fdist.
Simultaneously applying two forcing functions requires a method that allows distinguishing their contribution in the measurements. A well-documented method consists of assigning different sets of discrete points in the available
frequency range to Fdist andxtar [31], [32]. Fig. 2a shows the frequency separation of the two forcing signals
used in this research. Each set of frequency points is composed of a cluster of two adjacent frequencies to allow for frequency averaging during estimation.
For both forcing functions, the phases ψi were chosen randomly to obtain unpredictable behavior. A cresting
technique was applied to avoid peaks in the time domain [30], [33]. This technique minimizes the Cresting Factor (CF), which is defined as the maximum amplitude of the signal divided by the Root Mean Square (RMS) of the signal. By minimizing the cresting factor, the power of the signal can be higher without increasing the signal amplitude. The increased signal power generally results in better Signal-to-Noise Ratios (SNR), and, consequently, in better estimates.
The choice of signal amplitudes must account for human limitations. For the target positionxtar, the sinusoidal
amplitudes at high frequencies should not be too large, since the pilot must be able to follow them. The amplitude
distribution was chosen to match the frequency response of a filterHf [25]:
Hf =
(s + 10)2
(s + 1.25)2 . (3)
The amplitudes for Fdist were tuned to reach a trade-off between two conflicting objectives: they have to be
low enough not to disturb pilots during the tracking task, but still allow distinguishing their contribution in the measurements. Furthermore, the “Reduced Power Method” was used [29], see Fig. 2a. The principle of this method is to apply a reduced power level at higher frequencies, to allow estimation in a wide range of frequencies without influencing the control behavior of pilots.
The multisine time signals were obtained from their spectra using the inverse Fast Fourier Transform [30]. Table I
lists the base-frequency multiple k, the actual frequency f , the amplitude A and the phase φ of each of the sines
B. Cross-Spectral Density Analysis
A common method for identification of control behavior is based on cross-spectral density estimates. This approach, referred to as the CSD-based method, has previously been used for pilots in a pitch tracking task [23], [26] and for car drivers in a car following task [31].
In a closed multi-loop system like in Fig. 1, the best linear approximation of the pilot admittance can be calculated as [21], [33]: ˆ HADM(f ) = ˆ SFdistxCD(f ) ˆ SFdistFadm(f ) , f ∈ {fd} (4)
where{fd} is the set of frequencies in which Fdist has power, ˆSFdistxCD and ˆSFdistFadm are the estimated
cross-spectral densities betweenFdistandxCD and betweenFdistandFadm, respectively.
The CSD-based method assumes linearity of the pilot admittance; the validity of this assumption can be checked with the squared coherence function defined as [21]:
ˆ Γ2(f ) = | ˆSFdistxCD(f )| 2 ˆ SFdistFdist(f ) ˆSxCDxCD(f ) , f ∈ {fd} . (5)
The coherence function ˆΓ equals 1.0 when the system is linear and there is no noise, and approaches 0.0 when no
linear relation is found.
To estimate the cross-spectral density ˆSFdistFadm in (4), the time realization of the forceFadm must be known.
Unfortunately, only the total human forceFhuman on the control device can be measured, which is given by the
sum ofFadm andFvis (and the remnant noisen). However, if it is assumed that the visual response Fvis does not
contain power at frequencies in {fd}, the force Fadm at frequencies {fd} can be approximated with the human
forceFhumanmeasured at frequencies{fd}. In this case, the admittance at frequencies {fd} can be estimated given
measurements ofxCD,FdistandFhumanas:
ˆ HADM(f ) = ˆ SFdistxCD(f ) ˆ SFdistFadm(f ) ≈ SˆFdistxCD(f ) ˆ SFdistFhuman(f ) , f ∈ {fd} . (6)
Note that only if the power of the tracking errore is zero for all frequencies{fd} does Fvis not contain power
at frequencies in{fd}. This is because the visual response VIS in Fig. 1 is considered linear. Therefore, to apply
the common identification method, it must be assumed that the power of the tracking errore must be zero for all
frequencies{fd}: the non-interference assumption.
When the non-interference assumption is not fulfilled and Fadm cannot be approximated with Fhuman, the
estimator in (6) actually estimates a transfer function different from the admittance. The estimated Frequency Response Function (FRF) can be written as:
ˆ HADM(f ) = ˆ SFdistxCD(f ) ˆ SFdistFhuman(f ) = ˆ SFdistxCD(f ) , ˆ SFdistFdist(f ) ˆ SFdistFhuman(f ) , ˆ SFdistFdist(f ) = HˆFdistxCD(f ) ˆ HFdistFhuman(f ) , f ∈ {fd} (7)
where ˆHFdistxCD is the FRF from Fdist to xCD, and ˆHFdistFhuman is the FRF from Fdist to Fhuman. Note that
pilot model in Fig. 1, an analytical expression ofHFdistFhuman is obtained as a function of the transfer functions
HADM,HV IS,HCE, andHCD, i.e.:
HFdistFhuman = −HCD(HCEHV IS+ HADM−1 ) 1 + HCD(HCEHV IS+ HADM−1 ) =−H HCDHCEHV ISHADM + HCD ADM + HCDHCEHV ISHADM+ HCD . (8)
Similar steps are applied to find an analytical expression for the transfer functionHFdistxCD:
HFdistxCD = HCDHADM HADM+HCD 1 + HCDHADM HADM+HCDHV ISHCE = HCDHADM HADM + HCD+ HCDHADMHV ISHCE . (9)
More details on obtaining the previous expressions is given in the Appendix. Dividing (9) by (8), an analytical expression of the estimated admittance is found:
ˆ HADM = HFdistxCD HFdistFhuman = HCDHADM HCDHCEHV ISHADM+ HCD = HADM 1 + HCEHV ISHADM . (10)
It is clear that (6) represents a biased estimate of the admittance. Only at those frequencies where the product of
HV IS,HCE, andHADM is small compared to 1 (HCEHV ISHADM 1), ˆHADM approaches the real admittance
HADM. Therefore, if the non-interference assumption is violated, the estimated admittance may become unreliable.
Another drawback of the CSD-based method is that the pilot visual responseHV IS can not be obtained directly,
because the forceFviscan not be measured. On the contrary, the transfer function between the visual errore and the
position of the controlled element xCE, i.e., the open-loop transfer function, can be estimated using cross-spectra
estimates [21], [33]. In this case, the target forcing function xtar is considered as the external deterministic input
uncorrelated with the remnant noise. The pilot open-loop FRF can be estimated as: ˆ HOL(f ) = ˆ SxtarxCE(f ) ˆ Sxtare(f ) , f ∈ {ft} (11)
where{ft} is the set of frequencies in which xtar has power.
The coherence function corresponding to the open-loop transfer function is given by: ˆ Γ2OL(f ) = | ˆ SxtarxCE(f )| 2 ˆ Sxtarxtar(f ) ˆSxCExCE(f ) , f ∈ {ft} . (12)
IV. NOVELIDENTIFICATIONTECHNIQUES
To address the limitations of the conventional identification method described in the previous section, two novel identification techniques are presented. The first method operates in the time-domain and uses an AutoRegressive model with eXogenous inputs (ARX) to fit measured data. The second identification method operates in the frequency domain and uses cross-spectral density estimates similarly to the conventional technique. However, it does not require the non-interference assumption.
A. AutoRegressive models with eXogenous inputs
Linear Time-Invariant (LTI) polynomial models, such as the ARX, ARMAX, and Box-Jenkins (BJ) models, are commonly used for the identification of a large variety of dynamic systems [34]. These models describe the relationship between the system inputs, the noise and the system outputs with parametric rational transfer functions.
Based on time measurements of input and output signals of a generic dynamic system, an optimization procedure is used to find the LTI polynomial model that fits the time signals best.
Polynomial models were previously used to estimate human admittance during a car following task [31]. The driver model consisted of neuromuscular and visual responses as shown in Fig. 1. To identify the neuromuscular
response, the response to the visual errore was treated as an external filtered Gaussian noise. Single-input BJ models
were then used to estimate the neuromuscular response. Although this approach provided reliable neuromuscular
estimate [31], it may fail when the response to the visual errore cannot be approximated as an external Gaussian
noise.
The ARX method described in our paper explicitly accounts for the response to the visual error e by using a
multi-input ARX method. The stick deflection xCD can be considered as the output of a linear system that has
a visual errore and a disturbance force Fdist as inputs, and that is perturbed by the remnant noise n (Fig. 1). A
multi-input ARX model is then used to describe this dynamic system [34]:
xCD(t) = He(q)e(t) + Hf(q)Fdist(t) + Hn(q)n(t) = Be(q) A(q)e(t) + Bf(q) A(q) Fdist(t) + 1 A(q)N (t) , (13) with: A(q) = 1 + a1q−1+ . . . + anaq −na (14)
Be,f(q) = b0e,f+ b1e,fq
−1+ . . . + b nbe,bfq
−nnbe,bf
. (15)
He andHf represent the transfer functions from e and Fdist toxCD, respectively, which are related to the visual
responseHV IS and the admittanceHADM. The transfer functionHn models the effect of the remnant noisen on
stick deflection. In the ARX model, this effect is represented by the white noise signalN (t) filtered by the transfer
function A(q)1 . The terms na, nbe, and nbf are the order of the corresponding polynomials. It should be noted
that all the transfer functions of the system have the same set of poles. Although this coupling may be unrealistic, simulations and experimental results in sections V and VI will show that this does not to affect the identification accuracy.
Given the measurements ofxCD,Fdist, ande, the polynomial orders and the parameter values of the ARX model
are estimated as the optimal solution of a suitable cost function. In this paper, the Akaike final prediction error was used as the cost function [34]. The optimal least-squares solution comes in closed form as a linear regression, and does not suffer from local minima. It should be noted that this is not the case for other polynomial model structures like ARMAX and BJ [34].
No requirements need to be imposed on frequency separation between the inputs signalse and Fdist in order
to obtain reliable estimates [25]. Thus, contrary to the CSD-based method, the ARX method does not require any
assumption about possible interference of the feedback fromFdistwith the visual error e.
diagram algebra: ˆ HADM = HCDHf HCD− Hf (16) ˆ HV IS= He Hf (17) ˆ HOL= HeHCE . (18)
An overall validity of the ARX models is assessed by the Variance Accounted For (VAF), which shows how well the model can predict the measured output signal [25]. The VAF is defined as:
V AF = 1− N P k=1|x CD(tk)− ˆxCD,ARX(tk)|2 N P k=1|x CD(tk)|2 × 100% (19)
wherexCD(tk) and ˆxCD,ARX(tk) represent the measured and the predicted stick deflection, respectively (k indexes
the time samples). The value of the VAF varies between0% and 100%, where 100% indicates that the ARX model
completely describes the measured system response. Lower values are an indication of a worse model fit due to noise, nonlinearities, or unmodeled system characteristics.
B. Multi-loop cross-spectral densities
This section presents a second novel identification method based on cross-spectral density estimates, which explicitly accounts for the multi-loop nature of the considered control task. This method, referred to as the
CSD-ML method, does not require the non-interference hypothesis on the power content of the visual errore and does
not assume a rigid predefined model structure as the ARX approach.
In the CSD-ML method, the cross-spectral density analysis is used to estimate the following auxiliary FRFs:
• the FRF of the transfer functionHxtarxCD from the target positionxtar to the CD deflection xCD, i.e.:
ˆ HxtarxCD(f ) = ˆ SxtarxCD(f ) ˆ Sxtarxtar(f ) , f ∈ {ft} . (20)
• the FRF of the transfer function HFdistxCD from the force disturbance Fdist to the CD deflection xCD, i.e.,
the so-called force disturbance feedthrough [35]: ˆ HFdistxCD(f ) = ˆ SFdistxCD(f ) ˆ SFdistFdist(f ) , f ∈ {fd} . (21) ˆ
HxtarxCD and ˆHFdistxCD represent closed-loop dynamics from xtar andFdist toxCD, respectively. Therefore
they include dynamics of admittanceHADM, visual response HV IS, controlled element HCE, and control device
HCD, see Fig. 2. The feedback loop fromFdistto the visual errore is explicitly accounted for in the closed-loop
dynamics. Thus, the presence of any frequency content originating fromFdistin the visual error does not represent
a problem, as its effect on pilot behavior is explicitly accounted for.
The admittance ˆHADM and visual response ˆHV IS are computed from ˆHxtarxCD and ˆHFdistxCD using block
frequencies{ ¯f} over which they can be combined. Unfortunately, ˆHxtarxCD and ˆHFdistxCD are known only for the
sets{ft} and {fd} respectively, which do not have frequency points in common. However, since it is reasonable to
assume a “smooth” behavior of the frequency response functions ˆHxtarxCD and ˆHFdistxCD, a linear interpolation
between neighboring frequencies can be applied to obtain ˆHxtarxCD and ˆHFdistxCD on frequencies at which they
are not known. The CSD-ML method uses interpolation to estimate ˆHxtarxCD in the set {fd} and ˆHFdistxCD in
the set{ft}. A similar interpolation approach was used in [35].
It should be noted that ˆHxtarxCD can only be interpolated on a limited subset of {fd}, due to the design of
disturbance forcing functions. As shown in Fig. 2, the frequency range of{fd} (0.05 Hz - 10 Hz) is larger than
the frequency range of{ft} (0.1 Hz - 3 Hz). Since an interpolation of ˆHxtarxCD outside the range of{ft} would
produce unreliable estimates, the interpolation procedure can only be applied to the frequencies of{fd} in the range
0.1 Hz - 3 Hz. The resulting set { ¯f}, in which both ˆHxtarxCD and ˆHFdistxCD could be estimated, contained all
frequencies in{fd} and {ft} in the range 0.1 Hz - 3 Hz.
Analytical expressions for HFdistxCD and HxtarxCD can be given as functions of the various system transfer
functions, see equations (9) and (22):
HxtarxCD =
HV ISHHADMCDH+HADMCD
1 + HCDHADM
HADM+HCDHV ISHCE
. (22)
The derivation of these expressions is detailed in the Appendix. The FRFs of pilot admittance and visual response are estimated from (9) and (22) by using the estimated ˆHxtarxCD and ˆHFdistxCD:
ˆ HV IS(f ) = ˆ HxtarxCD(f ) ˆ HFdistxCD(f ) (23) ˆ HADM(f ) = ˆ HFdistxCD(f )HCD(f ) HCD(f )− HCD(f ) ˆHxtarxCD(f )HCE(f )− ˆHFdistxCD(f ) . (24)
Each FRF is calculated at the set of frequency points{ ¯f} resulting from the interpolation. In addition, the open-loop
FRF ˆHOL from the visual errore to the CE position xCE is calculated as:
ˆ HOL(f ) = HCE(f ) ˆ HxtarxCD(f ) 1− ˆHxtarxCD(f )HCE(f ) , f ∈ { ¯f} . (25) V. OFF-LINE SIMULATIONS
This section presents a comparison of the three identification techniques introduced in sections III and IV. A Monte-Carlo simulation of the multi-loop pilot model with a fixed remnant power was used to validate the identification methods. In addition, the effect of the controlled element gain on the accuracy of admittance estimation was assessed. Finally, the identification methods were tested with increasing levels of the remnant power in order to analyze their robustness.
A. Simulations with reference parameter values
The model in Fig. 1 was simulated with 100 different realizations of the remnant noisen, obtained as a Gaussian
white noise filtered by a third-order-low-pass filter Hn [36]:
Hn(s) = KN
12.73
10−1 100 101 10−10 10−5 Frequency [Hz] S ee [rad 2/Hz] e f d f t
Fig. 3. A typical power spectral density plot of the simulated visual error e.
The value of the gain KN was set to 2.12, in order to obtain a ratio of 0.5 between the remnant power and the
power of the linear part of the pilot response: Fvis− Fadm.
All simulations used fixed dynamics for the dynamic elements in Fig. 1. The CE dynamics were modeled similar to a previous study [37]:
HCE(s) = 18
s(s + 3) [rad/rad] . (27)
The CD dynamics were based on the identified response of a control device from Wittenstein Aerospace & Simulation GmbH, Germany:
HCD(s) = 1
1.522s2+ 8.832s + 86.469 [rad/N ] . (28)
The parameters for the pilot model were derived from Damveld et al. [23] by adapting their values to match the measurements from experimental evaluations (section VI). Based on [23], the visual response was modeled by a gain, a lead-lag filter and a time delay:
HV IS(s) = Kv
1 + sTlead
1 + sTlag
e−sτv [N/rad] (29)
whereKv = 20 N/rad, Tlead = 0.3 s, Tlag = 0.04s, and τv = 0.2701 s. The model of the arm admittance was
determined heuristically to fit the non-parametric estimation shown in [23]. The resulting transfer function is given by:
HADM(s) =
4.566· 10−6s3+ 0.0046s2+ 1.333s + 97.52
s3+ 82s2+ 712.2s + 1.167· 10−4 [rad/N ] . (30)
Fig. 3 shows a typical Power Spectral Density (PSD) of the simulated visual error e. The power of e is not
negligible at frequencies{ft} and {fd} where xtar andFdisthave power, respectively. This means that the force
disturbance Fdist produces a large interference on the visual error e and the non-interference hypothesis is not
fulfilled. Therefore, the CSD-based method is expected to provide biased estimates of neuromuscular response. In contrast, ARX and CSD-ML methods are expected to provide reliable estimates.
10−1 100 101 10−3 10−2 Frequency [Hz] |HADM | [rad/N] Analytical CSD−based (a) CSD-based. 10−1 100 101 10−3 10−2 Frequency [Hz] |HADM | [rad/N] Analytical ARX (b) ARX. 10−1 100 101 10−3 10−2 Frequency [Hz] |HADM | [rad/N] Analytical CSD−ML (c) CSD-ML.
Fig. 4. Comparison between admittance estimates given by the three identification methods. The estimates are averaged over all the simulations, and the shaded areas show the (positive) standard deviation (mean + SD).
10−1 100 101 101 102 Frequency [Hz] |HVIS | [N/rad] Analytical ARX (a) ARX. 10−1 100 101 101 102 Frequency [Hz] |HVIS | [N/rad] Analytical CSD−ML (b) CSD-ML.
Fig. 5. Comparison between visual response estimates given by ARX and CSD-ML methods. The estimates are averaged over all the simulations, and the shaded areas show the (positive) standard deviation (mean + SD).
The admittance estimates given by the three identification methods are shown in Fig. 4. The CSD-based approach
does not give correct estimates of the admittance at low frequencies, where the power level ofe at the frequencies
in{fd} is comparable to the power level at the frequencies in {ft} (see Fig. 3). On the contrary, both ARX and
CSD-ML methods give good results. CSD-ML provides admittance estimate on a relatively small frequency range, due to interpolation.
The pilot visual response can only be directly estimated with the ARX and CSD-ML methods. Fig. 5 shows that both methods give reliable estimates. Finally, the estimated open-loop transfer function fits the simulated response well for all identification methods, see Fig. 6.
10−1 100 101 10−4 10−2 100 Frequency [Hz] |HOL | [rad/rad] Analytical CSD−based (a) CSD-based. 10−1 100 101 10−4 10−2 100 Frequency [Hz] |HOL | [rad/rad] Analytical ARX (b) ARX. 10−1 100 101 10−4 10−2 100 Frequency [Hz] |HOL | [rad/rad] Analytical CSD−ML (c) CSD-ML.
Fig. 6. Comparison between open-loop transfer function estimates given by the three identification methods. The estimates are averaged over all the simulations, and the shaded areas show the (positive) standard deviation (mean + SD).
B. Variation of the gain of the controlled element
As described in section III-B, admittance identified with the CSD-based method can be influenced to a large
extent by the power content of the visual error e. High power content in the visual error signal e at frequencies
{fd} results in a biased estimate. As shown in Eq. (10), the CSD-based method produces reliable estimates only at
frequencies whereHCEHV ISHADM 1. This was investigated by performing multiple simulations with different
values of the gain of the controlled element. It was hypothesized that the CSD-based method performs better with
small gains ofHCE, whereas higher gains would result in a biased estimate of the admittance.
The dynamics for CD, ADM, VIS were the same as in section V-A. The CE dynamics was modeled as:
HCE = KCE 3
s(s + 3) [rad/rad] . (31)
Five different values ofKCE were tested:0, 3, 6, 9, 12. The condition KCE= 0 represents an open-loop situation,
where the non-interference hypothesis is completely fulfilled. No additional noise was inserted into the loop, as this effect will be tested explicitly in the next section.
Fig. 7 shows the admittance estimated by the three methods. As expected, the performance of the CSD-based
method deteriorates with increasing values ofKCE. In contrast ARX and CSD-ML methods are not influenced by
changes in the KCE value.
It should be noted that, in a realistic scenario, pilots would adapt their behavior to the controlled element [38]. However, this section aimed to highlight the influence of non-interference hypothesis on the admittance estimation, and this effect becomes more clear when the pilot is considered invariant.
10−1 100 101 10−3 10−2 Frequency [Hz] |HADM | [rad/N] Analytical K CE increasing (a) CSD-based. 10−1 100 101 10−3 10−2 Frequency [Hz] |H ADM | [rad/N] Analytical Estimated (b) ARX. 10−1 100 101 10−3 10−2 Frequency [Hz] |H ADM | [rad/N] Analytical Estimated (c) CSD-ML
Fig. 7. Admittances estimated for different values of KCE(KCE= 0, 3, 6, 9, 12). The ARX and CSD-ML methods provide similar estimates for all values of KCE.
C. Variation of the power of noise
The robustness of the three identification methods to increasing levels of pilot remnant was investigated by performing multiple simulations of the closed-loop control task with increasing levels of remnant noise power. The
remnant gain KN in (26) was gradually increased, resulting in 7 ratios between remnant power σN2 and signal
powerσ2
F, see table II. For each remnant level,100 simulations were performed with different noise realizations.
The analytical dynamics for CE, CD, ADM, VIS were chosen as in section V-A.
The admittance, the visual response and the open-loop transfer function were estimated by applying the three
identification methods. To compare the performance of the methods, a cost functionJ was calculated, representing
the logarithmic mean squared error between the estimated FRF ( ˆH) and analytical FRF (H) [39]:
J = 1 N N X k=1 log ˆH(fk) H(fk) ! 2 . (32)
This cost function accounts for both relative errors in the magnitude, and absolute errors in the phase. In order to
obtain meaningful values, J was calculated considering the frequenciesfkthat are common to the three identification
methods, i.e., the frequencies in {fd} between 0.1 Hz and 3 Hz for admittance estimates, and the frequencies in
{ft} for estimates of visual and open loop transfer functions.
Fig. 8 shows how cost functionJ varies with respect to the power ratio σ2
N/σF2. For all the estimates, J increases
with increasing values of the remnant power, implying a growing discrepancy between analytical and estimated FRFs. The ARX method appears to be the most robust with respect to noise. This is due to the fact that the ARX method explicitly accounts for the remnant noise in the estimation procedure. Considering the admittance estimates,
0 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 σ N 2/σ F 2 [−] J CSD−based ARX CSD−ML (a) Admittance. 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 σ N 2/σ F 2 [−] J ARX CSD−ML (b) Visual response. 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 3.5 σ N 2/σ F 2 [−] J CSD−based ARX CSD−ML
(c) Open loop transfer function. Fig. 8. Averaged values of the cost function J for different remnant levels.
the CSD-based method also shows a non-zero value of J when the remnant noise has zero power, i.e.,σ2
N/σ2F = 0
in Fig. 8a, indicating how the CSD-based method also provides a biased admittance estimate without remnant noise. Considering the visual response estimates, CSD-ML and ARX methods show a similar degradation of accuracy. Also for the open-loop transfer function estimates, the three methods show comparable degradation trends.
VI. EXPERIMENTAL EVALUATIONS
A. Method
The three identifications methods were applied to data obtained from an experimental manual control task. The experiment was performed using a control-loaded cyclic stick from Wittenstein Aerospace & Simulation GmbH, Germany, whose identified response dynamics are given in Eq. (28).
Eight participants performed a compensatory pitch tracking task and the roll axis of the stick was fixed. Before starting the experiment, participants received an overview about the scope of the experiment, and were instructed to minimize the tracking error shown on a primary flight display. The control device was continuously perturbed by a disturbance force. Both the reference signal and the disturbance force were designed as described in section III-A.
After a training phase, participants performed8 experimental trials. Data were logged at 100 Hz and the last 213
samples (= 81.92 s) of each trial of 90 s were used for identification purposes. Before identification, the data of
the8 trials performed by each participant were averaged in the time domain to reduce the effect of remnant noise.
Fig. 9 illustrates the PSD of the error signal e, the human force Fhuman, the control signal xCD, and the pitch
anglexCE for a single participant. The Signal-to-Noise Ratios (SNR) were high for most of the input frequencies
10−1 100 101 10−10 10−5 100 Frequency [Hz] Sff [N 2/Hz] F hum f d f t
(a) Human force Fhuman.
10−1 100 101 10−15 10−10 10−5 Frequency [Hz] Sxx [rad 2/Hz] x CD f d f t
(b) Control device deflection xCD.
10−1 100 101 10−15 10−10 10−5 Frequency [Hz] See [rad 2/Hz] e f d f t (c) Visual error e. 10−1 100 101 10−15 10−10 10−5 Frequency [Hz] Sxx [rad 2/Hz] x CE f d f t (d) Pitch angle xCE.
Fig. 9. Power spectral densities. The crosses and the points represent the disturbance force and target position frequencies, respectively.
The identification methods described in sections III and IV were then applied to the measured data. It was hypothesized that the CSD-based method would be influenced by the non-interference assumption, leading to estimates of admittance that differ from the ones obtained with the ARX and the CSD-ML methods. Furthermore, it was expected that the ARX and the CSD-ML provide similar admittance estimates.
B. Results
A sample PSD of the visual error is depicted in Fig. 9c. The power ofe contained large peaks at frequencies{fd}
where the force disturbance Fdist has power, especially at low frequencies. This indicates thatFdist had a large
influence on the visual errore and the non-interference assumption of the CSD-based method was not fulfilled.
Fig. 10 shows the admittance identified by the three identification methods. The CSD-based method gives the
admittance FRFs for all frequencies of{fd}, while the CSD-ML method only provides admittance estimates below
3 Hz, which is the highest value in the frequency set{ft} as discussed in section IV-B. On the other hand, the ARX
method provides a mathematical expression for the admittance transfer function, and the FRFs can be calculated for all the desired frequency points ({fd} ∪ {ft} in Fig. 10).
As expected, the admittance estimates given by ARX and ML were different from those given by the CSD-based technique, especially at lower frequencies. The admittance estimated by the CSD-method showed the same bias as found in simulations, see section V-A. At low frequencies (f < 0.2 Hz), the magnitude of the admittance estimated by CSD-based technique was lower than the magnitude given by the other two methods; the opposite occurred at medium frequencies (0.2 Hz < f < 2 Hz); at higher frequencies the differences between methods disappeared. On the other hand, the ARX and CSD-ML methods provided similar estimates.
10−1 100 101 10−4 10−3 10−2 Frequency [Hz] |H adm | [rad/N] CSD−based ARX CSD−ML
Fig. 10. Admittance frequency response functions for the three identification methods. The estimates are averaged over all participants.
10−1 100 101 100 Frequency [Hz] |Hden | [−] ARX CSD−ML (a) 10−1 100 101 10−4 10−3 10−2 Frequency [Hz] |Hadm | [rad/N] CSD−based ARX CSD−based corrected (b)
Fig. 11. (a) Bias Hdenof the admittance estimate by the CSD-based method. The bias is calculated using the FRFs estimated by the ARX and the CSD-ML methods. (b) Comparison between the admittance estimate by the CSD-based method and HADM/Hdenestimate by the ARX and the CSD-ML methods. The dynamics HADM/Hdenrepresent the biased admittance actually estimated by the CSD-based method.
Although the CSD-based method provided a biased admittance estimate, high squared coherence values were
found at all frequencies (Γ2> 0.8). In fact, the coherence function does not give information about the
correspon-dence between the actual and the estimated admittance, but only indicates that the relation between the measured
human forceFhumanand the position of the control devicexCD can be approximated as linear. The estimates given
by the ARX method were characterized by high VAF values (the mean value of the VAF was87%), indicating that
ARX models reproduce the measured signals well.
As shown in (10), the bias in the CSD-based method results from the admittance estimate including the dynamics
of the visual responseHV IS and the controlled elementHCE. The bias is negligible only at those frequencies where
Hden = 1− HV ISHCEHADM is close to 1, whereas it becomes larger whenHdendiverges from 1.Hdencan thus
be used as a measure of the bias of the CSD-based estimate. To evaluateHden, the actual dynamics of admittance
HADM and visual responsesHV ISare required. Since these dynamics are not known a priori, the estimates provided
10−1 100 102 103 Frequency [Hz] |H vis | [N/rad] ARX CSD−ML
Fig. 12. Comparison between visual responses given by ARX and CSD-ML methods.
frequencies above1 Hz, whereas it diverges from 1 at low frequencies. This is in complete agreement with results
shown in Fig. 10.
The bias measure Hden can be used to correct the biased admittance given by the CSD-based method. The
corrected admittance is obtained by multiplying the biased estimate byHden, see (10). Fig. 11b shows the biased
admittance given by CSD-based method, together with the corrected admittance and the ARX estimate. As the admittance estimate from the CSD-ML method was similar to the ARX estimate, it is not shown. As expected the corrected admittance is similar to the ARX estimate, confirming the validity of our analysis of the bias in the CSD-based method.
The pilot visual response can be computed directly only with ARX and CSD-ML methods. Fig. 12 shows that the estimates were comparable. The shape of the visual response resembled a gain at low frequencies and a differentiator at higher frequencies. The three identification methods gave comparable estimates also for the
open-loop transfer function HOL (Fig. 13). The open-loop responses resembled the dynamics of a single integrator at
frequencies below 1 Hz. This correlates fairly well with McRuer’s theories [14], which assess that pilots adapt
their responses to yield integrator-like dynamics of the open-loop transfer function around the crossover frequency
whereHOL= 100 rad/rad. The peak of the open-loop response at higher frequencies is related to the neuromuscular
dynamics [14].
VII. CONCLUSIONS
Three techniques for simultaneous identification of pilot neuromuscular and visual responses have been investi-gated. We showed that the identification method commonly used in literature (CSD-based method) may give biased estimates of admittance in cases when a so-called non-interference assumption is not fulfilled. Furthermore, we have derived an analytic expression for the bias in the CSD-based admittance estimate.
We presented two different procedures, one based on ARX models and a multi-loop cross-spectral density method, which allow overcoming this limitation. The results of offline simulations confirmed that both the ARX and the
10−1 100 10−1 100 Frequency [Hz] |H OL | [rad/rad] CSD−based ARX CSD−ML
Fig. 13. Comparison between open-loop transfer function estimates given by the three identification methods.
CSD-ML methods give reliable estimates also in the case when the non-interference assumption is not satisfied. The three identification methods were validated experimentally. The ARX and the CSD-ML methods showed estimates that were similar to each other, whereas the CSD-based method gave a biased estimate of admittance. Similar results were also found in simulation. The biased estimate of admittance from the CSD-based method could be corrected with experimental data, which corroborates our theoretical analysis of the CSD-based method.
Taken together, these results suggest that the admittance estimated by the CSD-based method cannot be considered to be reliable when the disturbance forcing function always has an influence on the visual error. On the contrary, the ARX and the CSD-ML methods are not influenced by the power content of the visual error.
The most important limitation of the ARX method lies in the fact that it assumes a rigid model structure that could be unrealistic. Despite this, the experimental results suggest that the ARX-model structure can describe our experimental data well. A number of potential limitations need to be considered also for the CSD-ML method. First, it gives admittance estimates in a reduced set of frequencies (at low frequencies with the forcing functions used in this paper). Furthermore, the interpolation procedure could yield noisy estimates. Nevertheless, we believe that this method could be a powerful tool to obtain an accurate low-frequency estimate when the influence of the disturbance force on the visual error is not negligible. The CSD-ML method neither makes a-priori hypothesis about the visual error, nor assumes a rigid model structure.
APPENDIX
In a closed-loop linear system as in Fig. 14a, the closed-loop transfer function Hcl between the input signalu
and the output signal y is given by:
Hcl=
H
1 + H· G (33)
where H and G represent the transfer functions of the system dynamics in the forward and feedback path,
H y
+−
u
G
(a) Generic closed-loop system. HCEHV IS − ++ N + Fhuman H−1 ADM − HCD Fdist
(b) Closed-loop system from Fdistto
Fhuman. HV IS HHADMCDH+HADMCD HCE + ++ xtar +− N + Fdist xCD
(c) Closed-loop systems from xtarand Fdist to xCD.
Fig. 14. Rearranged models of the compensatory tracking task in Fig. 1
The closed-loop dynamics between two generic signals of the compensatory tracking task in Fig. 1 can be obtained by rearranging the model to have a similar configuration as the system in Fig. 14a. Fig. 14b illustrates the
rearranged model to obtain the dynamics between the disturbance forceFdistand the human force Fhuman. Here,
the notationHSY S indicates the transfer function of SYS. The forward and feedback transfer functions are given
by:
H = HCD(−HCEHV IS− HADM−1 ), G =−1 . (34)
Fig. 14c shows the rearranged model to calculate the dynamics from xtar andFdistto xCD. The forward and
feedback transfer functions for the input signals xtar andFdistare:
xtar: H = HV IS HCDHADM HADM + HCD , G = HCE (35) Fdist: H = HCDHADM HADM+ HCD , G = HCEHV IS (36) ACKNOWLEDGMENTS
The work in this paper was partially supported by the myCopter project, funded by the European Commission under the 7th Framework Program. Heinrich H. B¨ulthoff was supported by the Brain Korea 21 PLUS Program through the National Research Foundation of Korea funded by the Ministry of Education. Correspondence should be directed to Heinrich H. B¨ulthoff.
REFERENCES
[1] T. M. Lam, M. Mulder, and M. M. van Paassen, “Haptic Interface in UAV Tele-Operation Using Force-Stiffness Feedback,” in Proc. IEEE Conf. Syst. Man Cybern., San Antonio, Texas, Oct. 2009, pp. 835 –840.
[2] S. Alaimo, L. Pollini, J.-P. Bresciani, and H. H. B¨ulthoff, “A Comparison of Direct and Indirect Haptic Aiding for Remotely Piloted Vehicles,” in Proc. 19th IEEE Ro-Man, Viareggio, Italy, Sep. 2010, pp. 506–512.
[3] T. Brandt, T. Sattel, and M. Bohm, “Combining Haptic Human-Machine Interaction with Predictive Path Planning for Lane-Keeping and Collision Avoidance Systems,” in IEEE Intell. Veh. Symp., Eindhoven, The Netherlands, Jun. 2007, pp. 582–587.
[4] K. Hashtrudi-Zaad and S. E. Salcudean, “Transparency in Time-Delayed Systems and the Effect of Local Force Feedback for Transparent Teleoperation,” IEEE J. Robot. Autom., vol. 18, no. 1, pp. 108–114, 2002.
[5] N. Parker, S. Salcudean, and P. Lawrence, “Application of Force Feedback to Heavy Duty Hydraulic Machines,” in Proc. IEEE Int. Conf. Robot. Autom., vol. 1, Atlanta, GA, May 1993, pp. 375–381.
[6] J. Rosen, B. Hannaford, M. MacFarlane, and M. Sinanan, “Force Controlled and Teleoperated Endoscopic Grasper for Minimally Invasive Surgery - Experimental Performance Evaluation,” IEEE Trans. Biomed. Eng., vol. 46, no. 10, pp. 1212–1221, 1999.
[7] G. P. Boucek Jr, J. E. Veitengruber, and W. D. Smith, “Aircraft Alerting Systems Criteria Study. Volume II. Human Factors Guidelines for Aircraft Alerting Systems,” Boeing Commercial Airplane Co., Seattle, WA, Tech. Rep. D6-44200, 1977.
[8] R. H. Cook, “An Automatic Stall Prevention Control for Supersonic Fighter Aircraft,” J. Aircraft, vol. 2, no. 3, pp. 171–175, May/Jun. 1965.
[9] D. A. Abbink and M. Mulder, “Exploring the Dimensions of Haptic Feedback Support in Manual Control,” J. of Computing and Information Science in Engineering, vol. 9, no. 1, pp. 011 006–1 – 011 006–9, Mar. 2009.
[10] L. Profumo, L. Pollini, and D. A. Abbink, “Direct and Indirect Haptic Aiding for Curve Negotiation,” in Proc. IEEE Conf. Syst. Man Cybern., Manchester, UK, Oct. 2013, pp. 1846–1852.
[11] K. Goodrich, P. Schutte, and R. Williams, “Haptic-Multimodal Flight Control System Update,” in Proc. 11th AIAA Aviation Technol., Integr., and Operations (ATIO) Conf., no. AIAA Paper 2011-6984, Virginia Beach, VA, Sep. 2011, pp. 1–17.
[12] S. De Stigter, M. Mulder, and M. M. Van Paassen, “Design and Evaluation of Haptic Flight Director.” J. Guidance Control Dyn., vol. 30, no. 1, pp. 35–46, Jan. 2007.
[13] D. A. Abbink, M. Mulder, F. C. T. van der Helm, M. Mulder, and E. R. Boer, “Measuring Neuromuscular Control Dynamics During Car Following with Continuous Haptic Feedback,” IEEE Trans. Syst., Man, Cybern. B, vol. 41, no. 5, pp. 1239–1249, 2011.
[14] D. T. McRuer, “Human pilot dynamics in compensatory systems: Theory, models and experiments with controlled element and forcing function variations,” Air Force Flight Dynamics Laboratory, Research and Technology Division, Air Force Systems Command, Wright-Patterson Air Force Base, OH, Tech. Rep. AFFDL-TR-65-15, 1965.
[15] D. T. McRuer and H. R. Jex, “A review of quasi-linear pilot models,” IEEE Trans. Hum. Factors Electron., vol. HFE-8, no. 3, pp. 231 – 249, Sep. 1967.
[16] R. A. Hess, “Pursuit Tracking and Higher Levels of Skill Development in the Human Pilot,” IEEE Trans. Syst., Man, Cybern., vol. 11, no. 4, pp. 262–273, Apr. 1981.
[17] D. T. McRuer, I. L. Ashkenas, and H. R. Pass, “Analysis of multiloop vehicular control systems,” Defense Technical Information Center, Wright-Patterson Air Force Base, OH, Tech. Rep. ASD-TDR-62-1014, 1964.
[18] R. L. Stapleford, D. T. McRuer, and R. E. Magdaleno, “Pilot Describing Function Measurements in a Multiloop Task,” IEEE Transactions on Human Factors in Electronics, vol. 8, no. 2, pp. 113–125, Jun. 1967.
[19] M. M. van Paassen, “Biophysics in Aircraft Control, A Model of the Neuromuscular System of the Pilot’s Arm,” Ph.D. dissertation, Delft Univ. of Technol., Delft, The Netherlands, 1994.
[20] M. M. van Paassen and M. Mulder, “Identification of Human Operator Control Behaviour in Multiple-Loop Tracking Tasks,” in Proc. 7th IFAC/IFIP/IFORS/IEA Symp. Anal. Design, Eval. Human-Mach. Syst., Kyoto, Japan, 1998, pp. 515–520.
[21] F. C. T. van der Helm, A. C. Schouten, E. de Vlugt, and G. G. Brouwn, “Identification of Intrinsic and Reflexive Components of Human Arm Dynamics during Postural Control,” J. Neurosci. Methods, vol. 119, no. 1, pp. 1 – 14, May 2002.
[22] D. A. Abbink, “Task Instruction: The Largest Influence on Human Operator Motion Control Dynamics,” in Proc. Sec. Joint EuroHaptics Conf. and Symp. on Haptic Interfaces for Virt. Environment and Teleoperator Systems, Tsukuba, Japan, Mar. 2007, pp. 206 –211. [23] H. J. Damveld, D. A. Abbink, M. Mulder, M. Mulder, M. M. van Paassen, F. C. T. van der Helm, and R. J. A. W. Hosman, “Identification
of the Feedback Component of the Neuromuscular System in a Pitch Control Task,” in Proc. AIAA Guidance Navigation Control Conf., no. AIAA 2010-7915, Toronto, Ontario Canada, Aug. 2010, pp. 1–22.
[24] M. Olivari, F. M. Nieuwenhuizen, J. Venrooij, H. H. B¨ulthoff, and L. Pollini, “Multi-loop Pilot Behaviour Identification in Response to Simultaneous Visual and Haptic Stimuli,” in Proc. AIAA Model. and Simul. Technol. Conf., no. AIAA 2012-4795, Minneapolis, Minnesota, Aug. 2012, pp. 1–23.
[25] F. M. Nieuwenhuizen, P. M. T. Zaal, M. Mulder, M. M. van Paassen, and J. A. Mulder, “Modeling Human Multichannel Perception and Control Using Linear Time-Invariant Models,” J. Guidance Control Dyn., vol. 31, no. 4, pp. 999–1013, Jul./Aug. 2008.
[26] H. J. Damveld, D. A. Abbink, M. Mulder, M. Mulder, M. M. van Paassen, F. C. T. van der Helm, and R. J. A. W. Hosman, “Measuring the Conribution of the Neuromuscular System During a Pitch Control Task,” in Proc. AIAA Model. and Simul. Technol. Conf., no. AIAA-2009-5824, Chicago, Illinois, Aug. 2009, pp. 1–19.
[27] D. A. Abbink, E. R. Boer, and M. Mulder, “Motivation for Continuous Haptic Gas Pedal Feedback to Support Car Following,” in IEEE Intell. Veh. Symp., Jun. 2008, pp. 283–290.
[28] A. C. Schouten, E. de Vlugt, B. J. van Hilten, and F. C. T. van der Helm, “Quantifying Proprioceptive Reflexes During Position Control of the Human Arm,” IEEE Trans. Biomed. Eng., vol. 55, no. 1, pp. 311 –321, Jan. 2008.
[29] W. Mugge, D. A. Abbink, and F. C. T. van der Helm, “Reduced Power Method: How to Evoke Low-Bandwidth Behaviour while Estimating Full-Bandwidth Dynamics,” in Proc. IEEE 10th Int. Conf. Rehabilitation Robot. ICORR, Noordwijk, The Netherlands, Jun. 2007, pp. 575– 581.
[30] A. C. Schouten, E. de Vlugt, and F. C. T. van der Helm, “Design of Perturbation Signals for the Estimation of Proprioceptive Reflexes.” IEEE Trans. Biomed. Eng., vol. 55, no. 5, pp. 1612–9, 2008.
[31] D. A. Abbink, “Neuromuscular Analysis of Haptic Gas Pedal Feedback during Car Following,” Ph.D. dissertation, Delft Univ. of Technol., Delft, The Netherlands, Dec. 2006.
[32] J. Venrooij, D. A. Abbink, M. Mulder, and M. M. van Paassen, “A Method to Measure the Relationship Between Biodynamic Feedthrough and Neuromuscular Admittance,” IEEE Trans. Syst., Man, Cybern. B, vol. 41, no. 4, pp. 1158 –1169, Aug. 2011.
[33] R. Pintelon and J. Schoukens, System Identification: a Frequency Domain Approach. Piscataway, NJ: IEEE Press, 2001. [34] L. Ljung, System Identification: Theory for the User, 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 1999.
[35] J. Venrooij, M. Mulder, D. A. Abbink, M. M. van Paassen, F. C. van der Helm, H. H. Bulthoff, and M. Mulder, “A New View on Biodynamic Feedthrough Analysis: Unifying the Effects on Forces and Positions,” IEEE Trans. Cybern., vol. 43, no. 1, pp. 129–142, Feb. 2013.
[36] P. M. T. Zaal, D. M. Pool, Q. P. Chu, M. M. van Paassen, M. Mulder, and J. A. Mulder, “Modeling Human Multimodal Perception and Control Using Genetic Maximum Likelihood Estimation,” J. Guidance Control Dyn., vol. 32, no. 4, pp. 1089–1099, 2009.
[37] R. A. Hess, “Modeling the Effects of Display Quality upon Human Pilot Dynamics and Perceived Vehicle Handling Qualities,” IEEE Trans. Syst., Man, Cybern., vol. 25, no. 2, pp. 338 –344, Feb. 1995.
[38] D. T. McRuer and D. H. Weir, “Theory of manual vehicular control,” Ergonomics, vol. 12, no. 4, pp. 599–633, 1969.
[39] R. Pintelon, P. Guillaume, Y. Rolain, J. Schoukens, and H. Van Hamme, “Parametric Identification of Transfer Functions in the Frequency Domain - a Survey,” IEEE Trans. Autom. Control, vol. 39, no. 11, pp. 2245 –2260, Nov. 1994.
[40] A. van Lunteren, “Identification of Human Operator Describing Function Models with One or Two Inputs in Closed Loop Systems,” Ph.D. dissertation, Delft Univ. of Technol., Delft, The Netherlands, Jan. 1979.
Mario Olivari (S’14) received the M.Sc. degree in automation engineering (cum laude) from the University of Pisa, Pisa, Italy, in October 2012, for his work on the identification of pilot control behavior in a closed-loop control task with haptic aids. Currently, he is a Ph.D student in the context of a collaborative project between the Department of Human Perception, Cognition and Action at the Max Planck Institute for Biological Cybernetics, T¨ubingen, Germany, and the Faculty of Automation Engineering at the University of Pisa.
His research interests include haptic guidance systems, pilot identification and the modeling of the neuromuscular system.
Frank M. Nieuwenhuizen received his M.Sc. degree in aerospace engineering from Delft University of Technology in 2005 for work on the identification of pilot control behavior in a closed-loop control task and its experimental application. In 2012, he received his Ph.D. degree for his research into the influence of simulator motion system characteristics on pilot control behavior, a collaboration between the Department of Human Perception, Cognition and Action at the Max Planck Institute for Biological Cybernetics and the Faculty of Aerospace Engineering at Delft University of Technology.
He is currently project leader for the myCopter project, which is funded by the European Commission under the Seventh Framework Programme.
Joost Venrooij (S10) received his M.Sc. and Ph.D. degrees (cum laude) in Aerospace Engineering from Delft University of Technology, The Netherlands, in August 2009 and March 2014 respectively, for his work on biodynamic feedthrough. Since December 2010 he is working in a collaborative project between the Delft University of Technology and the Max Planck Institute for Biological Cybernetics.
His research interests include biodynamic feedthrough, haptics, and motion perception.
Heinrich H. B ¨ulthoff (M’95) received his Ph.D. degree in natural sciences in 1980 from Eberhard Karls University, T¨ubingen, Germany.
From 1980 to 1985, he was Research Scientist with the Max Planck Institute (MPI) for Biological Cybernetics, T¨ubingen, and from 1985 to 1988 Visiting Scientist at the Massachusetts Institute of Technology, Cambridge. Between 1988 and 1993, he was Assistant Professor, Associate Professor and finally Full Professor of Cognitive Science at Brown University, Providence, RI. In 1993, he became the Director of the Department of Human Perception, Cognition and Action at the MPI for Biological Cybernetics, and scientific member of the Max Planck Society. Since 1996, he has been Honorary Professor with the Eberhard Karls University, and since 2009 Adjunct Professor with Korea University, Seoul, Korea.
His research interests include object recognition and categorization, perception and action in virtual environments, and human-robot interaction.
Lorenzo Pollini (M’93) received the “Laurea” degree in computer engineering (cum laude) and the Ph.D. degree in robotics and industrial automation from University of Pisa, Pisa, Italy, in 1997 and 2000, respectively.
From 2000 to 2002, he was an Instructor of Automatic Control with the Italian Navy Academy. He is currently an Assistant Professor of automatic control with the Faculty of Engineering, University of Pisa.
His research interests concern guidance and navigation systems, vision based control, haptic support systems, fuzzy and nonlinear adaptive control, and real time dynamic systems simulation with specific application to unmanned systems.
TABLE I
FREQUENCIES,AMPLITUDES,AND PHASES DEFINING THE DISTURBANCE AND TARGET FORCING FUNCTIONS.
Disturbance Fdist Target xtar
k f A φ k f A φ (-) (Hz) (N) (rad) (-) (Hz) (N) (rad) 4 0.0488 0.3810 1.3875 10 0.1221 0.0198 -0.7003 5 0.0610 0.3810 2.5511 11 0.1343 0.0198 2.2216 14 0.1709 0.3810 2.7917 20 0.2441 0.0111 4.5432 15 0.1831 0.3810 -0.4228 21 0.2563 0.0111 4.3424 24 0.2930 0.3810 -2.9281 30 0.3662 0.0065 4.5408 25 0.3052 0.3810 -1.8836 31 0.3784 0.0065 6.6869 34 0.4150 0.3810 -3.6960 40 0.4883 0.0042 4.6231 35 0.4272 0.3810 -4.1771 41 0.5005 0.0042 5.8999 44 0.5371 0.3810 -2.6694 50 0.6104 0.0030 8.3374 45 0.5493 0.3810 -3.5128 51 0.6226 0.0030 7.1502 54 0.6592 0.3810 -2.6805 60 0.7324 0.0023 7.4114 55 0.6714 0.3810 -0.1215 61 0.7446 0.0023 5.1712 64 0.7813 0.3810 2.9424 70 0.8545 0.0018 8.4636 65 0.7935 0.3810 2.8250 71 0.8667 0.0018 9.6620 74 0.9033 0.3810 -0.0129 80 0.9766 0.0015 8.6567 75 0.9155 0.3810 -1.2966 81 0.9888 0.0015 9.8456 94 1.1475 0.3302 2.2066 100 1.2207 0.0011 8.1844 95 1.1597 0.3302 2.0687 101 1.2329 0.0011 8.1368 124 1.5137 0.3302 0.1803 140 1.7090 0.0008 5.7817 125 1.5259 0.3302 0.4412 141 1.7212 0.0008 6.0687 154 1.8799 0.3302 3.8712 180 2.1973 0.0006 9.5528 155 1.8921 0.3302 3.7964 181 2.2095 0.0006 10.5519 194 2.3682 0.3302 0.4953 230 2.8076 0.0006 7.6383 195 2.3804 0.3302 1.6589 231 2.8198 0.0006 8.0429 254 3.1006 0.3302 2.1529 260 3.1738 0.0005 7.0004 255 3.1128 0.3302 -0.8407 261 3.1860 0.0005 6.3912 344 4.1992 0.3302 -0.2912 345 4.2114 0.3302 2.0218 484 5.9082 0.3302 4.7341 485 5.9204 0.3302 8.1897 694 8.4717 0.3302 5.8242 695 8.4839 0.3302 9.2787
TABLE II SIMULATED POWER RATIOS.
KN 0 2.21 3.5 4.6 6 7.6 10 σ2
N/σ
2
